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17th World Conference on Earthquake Engineering, 17WCEE
Sendai, Japan - September 13th to 18th 2020
Paper N° C001077
Registration Code: S-A00323
OPTIMAL GAIN CALIBRATION OF ADAPTIVE MODEL-BASED
COMPENSATION FOR REAL-TIME HYBRID SIMULATION TESTING
G. Fermandois(1), C. Galmez(2), M. Valdebenito(3)
(1) Assistant Profesor, Departamento de Obras Civiles – Universidad Técnica Federico Santa María, Santiago, Chile,
gaston.fermandois@usm.cl
(2) M.S. Student, Departamento de Obras Civiles – Universidad Técnica Federico Santa María, Santiago, Chile,
cristobal.galmez.12@sansano.usm.cl
(3) Associate Profesor, Departamento de Obras Civiles – Universidad Técnica Federico Santa María, Valparaíso, Chile,
marcos.valdebenito@usm.cl
Abstract
Real-time hybrid simulation (RTHS) is an experimental technique for structural testing, where a critical
element is studied in the laboratory, and the rest of the structure is represented through numerical simulations.
The boundary conditions on the physical specimen are imposed by a transfer system (i.e., actuators), and it is
essential to minimize synchronization errors between numerical and experimental subdomains to ensure not
only accurate but stable results during the experiment. Many control methods have been proposed to
compensate actuator dynamics and minimize synchronization errors. However, current methods require either
manual tuning of controller parameters, which is a time-consuming and challenging process. Alternatively,
model-based approaches require good knowledge of the transfer system (i.e., a calibrated model through
system identification), including any dynamic interactions with the physical specimen and surrounding loading
equipment. Thus, additional costs and sometimes premature damage to the physical specimen are expected
during the controller calibrations.
This paper conducts a detailed study on adaptive compensation for RTHS, with a focus on robustness against
the choice of adaptive gains. The main goal is to design a dynamic compensator that does not require previous
knowledge of the specimen interaction with the transfer system. Hence, an adaptive gain optimization
procedure is proposed to improve the robustness of this technique. Optimal adaptive gains are obtained through
a particle swarm optimization approach, where the evaluation of the objective function is carried out by a series
of numerical simulations of the interactions between specimen-transfer systems. As a proof-of-concept, this
method is evaluated using virtual RTHS simulations, including the controller design and calibration processes.
The method achieves excellent compensation using the same controller for various experimental scenarios,
including different partitioning cases, uncertainties in the actuator and experimental substructure parameters,
and noise in measured signals. Through this development, structural laboratories will be capable of testing
different substructures while avoiding unnecessary system identification to capture specimen interaction, and
without a significant compromise in accuracy or laboratory safety.
Keywords: Real-time hybrid simulation; dynamic compensation; adaptive control; adaptive gain calibration;
particle swarm optimization.
17th World Conference on Earthquake Engineering, 17WCEE
Sendai, Japan - September 13th to 18th 2020
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1. Introduction
Laboratory tests are essential to study the behavior of structural systems and materials subjected to dynamic
loadings, such as those produced by earthquakes. Experimental techniques such as cyclic tests and shake table
tests are quite ubiquitous in structural engineering. Another technique called real-time hybrid simulation
(RTHS) has proven as a cost-effective and realistic approach to conducting seismic performance assessment,
taking full advantage of available equipment installed in laboratories (e.g., dynamic actuators or shake tables).
Real-time hybrid simulation (RTHS) is an experimental technique for structural testing. A critical
element is studied in the laboratory, while the rest of the structure is represented through numerical simulations
[1,2]. At each time step, the calculated displacements are imposed on the experimental substructure by a
transfer system (i.e., actuators). Then, the restitutive forces are measured and incorporated into the equations
of motion to calculate the displacement at the following time step. Representing numerically part of the
structure reduces the costs and requirements of the laboratory, while the experimental part results in a realistic
analysis of the physical specimen.
A crucial aspect of RTHS tests is the correct application of the boundary conditions on the experimental
substructure. The dynamic properties of the transfer system and their interaction with the experimental
substructure produces synchronization errors (amplitude or delay errors) between commanded and measured
displacements [3]. The most harmful error is the delay between the physical and numerical domain. When this
is introduced in the equation of motion, it can cause not only precision problems but also instability [4]. Thus,
different methods have been proposed to compensate for the dynamic of the transfer system and thus minimize
synchronization errors. Early methods reported in the literature are based on polynomial extrapolation,
assuming a constant delay [5]. Other methods are based on representing the system to be controlled with a
first-order transfer function and applying the inverse of this function to compensate for the dynamic of the
actuator [6]. More sophisticated methods are known as model-based compensation, where the controller is
designed based on a higher-order estimate of the transfer function of the plant [7]. Model-based compensation
has excellent results if a good plant model is available to design the controller. However, there are no
guarantees of accuracy or stability when there is significant uncertainty on the model.
Adaptive compensation has been proposed in RTHS testing to provide excellent controller performance
when there is uncertainty or significant non-linearity in the control plant. Some adaptive methods consist of
compensation through a first-order transfer function, where parameters of the function are adjusted during the
test based on a frequency domain analysis of commanded and measured signals, such as Adaptive Phase-lead
Compensator [8] or Windowed FEI Compensation [9]. Other methods like Adaptive Time Series [10] and
Conditional Adaptive Time Series [11] estimate the plant through Taylor series expansion and adjust the
parameters in the time domain. Adaptive model-based compensation [12] consists of an estimate of the plant
in frequency-domain; then, compensation is implemented in time-domain using numeric derivatives of the
commanded signal and adaptation based on gradient. Some methods use polynomial extrapolation, such as the
Adaptive Two-Stage Compensation [13] or the Improved Adaptive Forward Prediction [14]. Usually, adaptive
methods have proved to be very efficient when there is uncertainty or non-linearity in the control plant;
however, they are highly dependent on prior knowledge of the system to establish initial conditions and
constraints for the adaptive parameters. Therefore, it is generally required to test the control plant with the
experimental substructure included in order to perform the design and calibration of the controller. This
methodology can cause premature damage to the physical specimen to be tested.
The objective of this study is to develop an adaptive compensator which is independent of the
experimental substructure to be tested, avoiding the premature testing of the physical specimen. The Adaptive
Model-based Compensation [12] method is used with the following modifications: (i) first-order controller;
(ii) modified filter for the adaptation process; (iii) initial parameters based on a model of the transfer system
without interaction with the experimental substructure; (iv) adaptive gains calibration through off-line
numerical simulations using particle swarm optimization. The method is implemented in a virtual RTHS
benchmark problem developed by Silva et al. [15]. This paper has the following organization. Section 2
describes the RTHS benchmark problem, the simulation cases, and the evaluation criteria to assess its
17th World Conference on Earthquake Engineering, 17WCEE
Sendai, Japan - September 13th to 18th 2020
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performance. Section 3 presents the proposed compensation method, with the formulation of the optimal
adaptive gain calibration process. Section 4 presents the main results of the simulations for different
partitioning cases in order to study the robustness of the proposed compensator. Lastly, Section 5 presents the
conclusions and final remarks.
2. Problem formulation
2.1 RTHS benchmark problem
The RTHS benchmark problem [15] consists of a three-story moment frame with three lateral degrees of
freedom, as shown on the left side of Fig. 1. The reference structure is divided into a numerical substructure
(NS) and an experimental substructure (ES), as shown on the right side of Fig. 1.
Fig. 1 – Reference structure and partitioning. Adapted from [15]. (Note: NS = dashed line; ES = red line.)
The equation of motion of the linear reference structure is presented in Eq (1).
(1)
where , , and are the mass, stiffness and damping matrices, respectively. , , and are the
displacement, velocity, and accelerations vectors, respectively, all measured relative to the ground motion.
is the ground acceleration, and is the seismic influence vector, taken as for this problem.
The equation of motion of the numerical substructure (NS) is presented in Eq (2).
(2)
where subscript refers to numerical substructure (NS). Subsequently, is the measured restoring force from
the experimental substructure (ES) described in Eq (3).
(3)
where , , and the mass, stiffness, and damping parameters of the experimental substructure. In RTHS,
the displacement of the first DOF from NS, , is considered the target displacement . Meanwhile,
corresponds to the measured displacement of ES and in the ideal case , which satisfies compatibility
in the boundary between substructures.
The block diagram utilized to solve this problem is presented in Fig. 2, which was implemented in
Matlab and Simulink. For direct numerical integration, a 4th Order Runge-Kutta solver with a fixed time step
17th World Conference on Earthquake Engineering, 17WCEE
Sendai, Japan - September 13th to 18th 2020
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[sec] was considered. The output signal from the dynamic compensator is called command
displacement and is denoted with the variable . Noise has been added to measured signals to simulate
physical sensors. Also, the output of the reference structure is the displacement of the first floor and is denoted
as . This signal is utilized to compare with the results of the RTHS to assess its accuracy.
Fig. 2 – Simulink model for reference structure and RTHS closed-loop model.
The control plant (i.e., the actuator connected to ES) is modeled as shown in Fig. 3, where corresponds
to the Laplace variable, ; is the complex number, and is the circular frequency in [rad/s]. The
parameters of the transfer system model are listed in Table 1, including uncertainties in three parameters
represented by normally distributed and mutually independent random variables.
Fig. 3 – Control plant model. Adapted from [15].
Table 1 – Transfer system parameters.
Parameter
Nominal Value
Standard Deviation
Units
m-Pa/sec
m-Pa
1/sec
Nondimensional
1/sec
17th World Conference on Earthquake Engineering, 17WCEE
Sendai, Japan - September 13th to 18th 2020
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2.2 Simulation cases
The benchmark problem proposes different partitioning cases consisting of different mass and damping for the
reference structure, resulting in different numerical substructure but the same experimental substructure for
each case. In this paper, four cases are presented to evaluate the performance of the proposed control method,
including uncertainties in the stiffness represented by a normally distributed random variable. The values for
each case are listed in Table 2. Case I correspond to the most sensitive case from the benchmark problem,
while Case II corresponds to the less sensitive. Case III and IV include different experimental substructures.
For each case, the reference structure has the same mass per floor and the same damping for all modes.
Table 2 – Properties of simulation cases.
Reference Structure
Numerical Substructure
Case
Earthquake
Mass
[kg]
Frequencies
[Hz]
Damping
[%]
[kg]
[kg/sec]
(mean)
[N/m]
(std.)
[N/m]
I
El Centro 40%
1000
3.6 ; 16.0 ; 38.1
3
29.1
114.6
1.19
5
II
Kobe 40%
1000
3.6 ; 16.0 ; 38.1
5
29.1
114.6
1.19
5
III
Kobe 40%
1100
3.1 ; 13.5 ; 32.2
3
40
200
1.79
10
IV
El Centro 50%
800
4.0 ; 17.9 ; 42.6
5
20
100
0.82
8
2.3 Evaluation criteria
The results are evaluated through three performance indicators:
• Normalized root-mean-square error between the target displacement and the measured
displacement (Eq (4)). Measure the synchronization error.
(4)
• : Corresponds to the delay indicator obtained with the frequency evaluation index [16].
Measure the synchronization delay (in milliseconds) between and .
• Normalized root-mean-square error between the reference displacement and the
measured displacement (Eq (5)). Measure the error between the simulation and the
reference structure.
(5)
3. Methodology
3.1 Adaptive model-based compensation
The control method presented in this paper is based on the Adaptive Model-based Compensation (AMBC) [12].
The original method consists of a third-order adaptive feedforward combined with an LQG feedback regulator.
In this study, a first-order adaptive controller is presented to show the calibration process in detail. The LQG
is not considered for two reasons: (i) to demonstrate the compensation capacity of the adaptive feedforward;
and (ii) the LQG method requires good knowledge to design the controller.
The formulation consists of estimating the plant by a first-order transfer function without zeros, as shown
in Eq (6).
17th World Conference on Earthquake Engineering, 17WCEE
Sendai, Japan - September 13th to 18th 2020
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(6)
Then, the inverse of this transfer function is used to generate the command signal using the target
displacement as input, as shown in Eq (7).
(7)
The parameters and must be adjusted during the test to achieve good compensation. Therefore,
Eq (6) is reordered to obtain a relation between the commanded and measured displacements, as shown in Eq
(8).
(8)
In the original AMBC method, a low-pass filter is used to make proper transfer functions to
obtain the derivatives of . This filter affects the amplitude of the measured signal deteriorating the
adaptation process. In this paper, a Butterworth filter is utilized to remove high-frequency noise and then
calculate a numeric derivative to obtain , just like in other methods such as Adaptive Time Series [10] and
Conditional Adaptive Time Series [11]. The filter is applied to the commanded signal to synchronize with the
filtered measured signal. Once obtained the derivatives of , an estimation error could be obtained, as
shown in Eq (9).
(9)
where is the filtered commanded signal , is the estimate of , is a vector that contains ,
and is the filtered measured signal .
is the normalizing signal
; with
in this paper. Once the estimated error is formulated, a cost function is defined in Eq (10).
(10)
Finally, using the adaptive gradient law, the rate of change is obtained, as shown in Eq (11).
(11)
where is the adaptive gain matrix. In this paper, is assumed to be a diagonal matrix, where
. These adaptive gains are related to the rate of adaptation parameters, and its calibration is
discussed in Section 3.3.
Additionally, the original AMBC method considered the Routh’s stability criterion to provide constraints
for the adaptive parameters . In this study, Routh’s stability criterion is considered insufficient because it is
based on the stability of the control plant being estimated, which does not guarantee the stability of the RTHS
close-loop system. Thus, the stability of a compensation method in RTHS still a pending issue.
3.2 Specimen interaction in RTHS testing
Due to control-structure interaction [3], the transfer function of the plant will change every time the
experimental substructure is changed, so in many control methods, the controller must be redesigned for each
particular case. In this paper, the controller is designed to be used with different experimental substructures,
avoiding system identification tests for each substructure to be tested.
The initial parameters and are defined with an estimation of the transfer system without
interaction with any experimental substructure. In this paper, the parameters and are only restricted to
and , allowing free adaptation. The filter utilized in adaptation law is a fourth-order Butterworth
17th World Conference on Earthquake Engineering, 17WCEE
Sendai, Japan - September 13th to 18th 2020
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filter with a cutoff frequency of 20 [Hz]. In Fig. 4 the transfer function associated with initial parameters
and are presented.
For the calibration process, a model of the transfer system connected to a calibration structure is utilized.
This structure must show considerable interaction with the transfer system, and it is used only to obtain a
calibration plant model from a commanded displacement to a measured displacement. In this paper, a
calibration structure different to the experimental substructures used in RTHS is utilized, whose properties are
[kg], [N/m] and [kg/sec]. The transfer functions of the calibration plant and
the plant for each RTHS case with nominal values are presented in Fig. 4. The transfer functions are presented
from 0 [Hz] to 100 [Hz], but the most significant frequencies in seismic testing are approximately 0-20 [Hz].
Cases I and II have the same experimental substructure, so present the same transfer function, while Case III
and Case IV exhibit different transfer functions due to the different experimental substructures. All the plants
have different behavior with respect to the initially estimated plant, showing different levels of interaction with
the transfer system. Thus, the parameters and must be adjusted to achieve good compensation.
Fig. 4 – Bode diagrams of transfer function
3.3 Optimal calibration of adaptive gains
For the calibration, a Simulink model is utilized to evaluate different off-line pairs of and . The Simulink
model is presented in Fig. 5 and consists in: (i) ground acceleration (in this calibration, El Centro earthquake
scaled to 60%); (ii) a calibration structure to generate a target displacement (in this calibration, a single degree
of freedom structure with natural frequency of 5 [Hz] and 5% damping ratio); (iii) the adaptive controller with
initial conditions according to red curve Fig. 4 and predefined adaptive gains and ; (iv) calibration plant
model (green curve of Fig. 4); and (v) noise addition to the measured displacement to model the sensor.
Fig. 5 – Simulink model for calibration.
17th World Conference on Earthquake Engineering, 17WCEE
Sendai, Japan - September 13th to 18th 2020
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The indicator is selected to evaluate the controller performance because it represents the tracking
error, and minimizing this error, stable and accurate results are expected. So, for each simulation, the
indicator is computed to measure the error between the target and measured displacement. Thus, the calibration
consists of obtaining the adaptive gains, which minimizes the performance indicator. This study proposes
particle swarm optimization to find the optimal adaptive gains, but other optimization methods could be used
as well. Particle swarm optimization [17] consists of defining several particles (), where the position of each
particle represents a possible solution to minimize a fitness function. In this calibration, represents the
position for the particle at the iteration , where is the total number of iterations.
The position is related to the adaptive gains such that . Thus, for each
position, there is an error indicator resulting from the simulation with the adaptive gains associated
with position . Next, a set of particles with random initial positions and velocities are defined in a
constrained search space. Then, the position and velocity of each particle are updated, according to Eq (12)
and Eq (13).
(12)
(13)
where is the velocity of particle at iteration . The velocity is updated according to a weighted sum of
three components: (i) inertia with weight ; (ii) difference between the position with best result of the particle
and the current position , with a random component and weight ; and (iii) the difference
between the position with best result of the swarm and the current position , with a random
component and weight . The best position of a particle corresponds to the position with the
lowest value for the particle . Whereas, the best position of the swarm corresponds to the position
with the lowest value for all particles.
In this calibration, the lower and upper bounds are set to and ,
resulting in and . The results of the optimization method in terms of value
for a pair [] are presented in Fig. 6 with a few iterations and particles ( and ; i.e.,
simulations) where each marker represents a particle and each color a different iteration. The contour map was
obtained from sampling with a uniform grid of 4 simulations. The contour levels show that many
combinations of ] result in good tracking (), while other combinations with higher values of
results in bad tracking performance (). Also, after analyzing the optimization process in Fig. 6, it
can be noticed that the particles move from their initial random position to the global optimal. Finally, the best
result of the swarm is indicated by the blue circle.
The adaptive gains obtained with different numbers of particles and iterations are presented in Table 3.
Finally, the best results are and . These adaptive gains are utilized for the
RTHS simulations, whose results are presented in Section 4.
Table 3 – Optimization results.
Optimization
No. Particles
No. Iterations
1
5.369
2
5.383
3
5.370
4
5.367
17th World Conference on Earthquake Engineering, 17WCEE
Sendai, Japan - September 13th to 18th 2020
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Fig. 6 – Particle swarm optimization for adaptive gains (Note: color = iteration; marker = particle).
4. Results
4.1 RTHS Case I
The results of RTHS Case I with nominal values and using and are presented
graphically in this subsection. In Fig. 7, the measured displacement is compared with the target displacement
for tracking evaluation. With [%] and [msec], good tracking is achieved, allowing a
stable test. Meanwhile, Fig. 8 shows a comparison between the measured displacement and the reference
displacement. The indicator reaches only 3.21%, which means accurate results were obtained.
Fig. 7 – Target and measured displacements results for Case I.
17th World Conference on Earthquake Engineering, 17WCEE
Sendai, Japan - September 13th to 18th 2020
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Fig. 8 – Reference and measured displacements results for Case I.
Parameter adaptation during the test is presented in Fig. 9, where it can be noted that the parameters show
more adaptation at the beginning of the earthquake (i.e., after 5 [sec]), which demonstrates fast adaptation. The
parameters show convergence during the earthquake, allowing to estimate the plant and achieving reasonable
compensation. The adaptation process for Case I is contrasted with adaptation for the other cases. The initial
parameters are the same for all cases but converge to different values due to the differences in the control plant.
The parameter is related to the amplitude errors of the control plant. Since both Cases I and II are subjected
to different excitations, the response of the plant in each case presents different amplitude errors. Thus,
parameter converges in each case to slightly different values. Meanwhile, the parameter is directly
related with the time delay of the control plant and can be observed that this parameter for both Cases I and II
converge to the same value, while Case III converge to higher value of and Case IV to a lower , which is
consistent with time delays presented in Fig. 4. Notice that parameters and try to identify the control
plant through a first-order transfer function. However, the real system has a more complex behavior, so the
estimated parameters may vary depending on the commanded displacements.
Fig. 9 – Parameter adaptation during the test for all cases.
17th World Conference on Earthquake Engineering, 17WCEE
Sendai, Japan - September 13th to 18th 2020
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4.2 Robustness of the proposed method
The statistics for 20 simulations for each case are presented in Fig. 10, where the red line corresponds to the
median, and the bottom and top limits indicate the 25th and 75th percentile, respectively. The whiskers extend
to the most extreme values. Outliers (if present) are represented by ‘+’ symbol. Notice that the same controller
allows excellent compensation for all different cases. This adaptation capacity allows not only to control
different plants, but it also shows excellent robustness against uncertainties and noise. It could also maintain a
suitable performance if the experimental substructure changes its properties considerably during the test.
Fig. 10 –RTHS performance with the proposed compensation method. (Note: 20 simulations per case.)
5. Conclusions
This study presents the design and calibration process of an adaptive controller for real-time hybrid simulation.
Initial parameters are based only on the transfer system without specimen interaction, and adaptive model-
based compensation is utilized to adjust control parameters during the test to capture the interaction above.
The adaptive gains of the controller are adjusted through “off-line” numerical simulations using particle swarm
optimization, which allows finding optimal gains with a reasonable number of iterations. After gain calibration,
a virtual RTHS experiment with different control plants was conducted, including uncertainties in the transfer
system and experimental substructure properties. The results demonstrate that a fixed robust controller could
be used with different experimental substructures, avoiding subsequent system identification tests. Even
though this study considered only linear systems for the numerical and experimental substructures, it is
expected that it also works with non-linear systems since the controller can adapt to different control plants.
The first-order control method presented in this paper could also be extended for a higher-order controller
using the same optimization process for the calibration, allowing to represent more complex control plants.
Finally, this adaptive method should be complemented with a stability analysis during the RTHS before its
application in the laboratory.
6. Acknowledgments
The authors gratefully acknowledge the financial support from the Universidad Técnica Federico Santa Maria
(Chile) through the research grant Proyecto Interno de Línea de Investigación No. PI_L_18_07, and from the
Chilean National Commission for Scientific and Technological Research (CONICYT), FONDECYT-
Iniciación research project No. 11190774. Any opinions, findings, conclusions, or recommendations expressed
in this paper are those of the authors and do not necessarily reflect those of the sponsors.
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